At room temperature the electrical conductivity of \(\mathrm{PbS}\) is \(25(\Omega-\mathrm{m})^{-1}\), whereas the electron and hole mobilities are 0.06 and \(0.02 \mathrm{m}^{2} / \mathrm{V}-\mathrm{s},\) respectively. Compute the in trinsic carrier concentration for PbS at room temperature.

Electrical conductivity is a measure of a material's ability to conduct an electric current. When we talk about conductivity in semiconductor materials like Lead Sulfide (PbS), it's crucial to understand the roles played by free charge carriers: electrons and holes. These carriers are mobile and can move through the material under the influence of an electric field, contributing to the overall conductivity.

Intrinsic semiconductor materials, such as pure silicon or germanium, have an equal number of free electrons and holes when undoped. The electrical conductivity, represented by the Greek letter \( \sigma \), is thus directly proportional to the product of the charge of an electron \( q \), the carrier concentration \( n \) (for electrons) or \( p \) (for holes), and their respective mobilities \( \mu_e \) and \( \mu_h \).

The formula \( \sigma = q * (n * \mu_e + p * \mu_h) \) sets the stage for determining intrinsic carrier concentration, which is a fundamental property describing the number of charge carriers in a semiconductor material under no external doping or excitation.

Mobility signifies how quickly an electron or a hole can move through a semiconductor under the influence of an electric field. The electron mobility \( \mu_e \) and hole mobility \( \mu_h \) are critical parameters that determine how rapidly a semiconductor can respond to electrical stimuli.

For example, in the exercise, the electron and hole mobilities for PbS are given as 0.06 and \(0.02 \mathrm{m}^{2} / \mathrm{V}-\mathrm{s}\), respectively. These values are essential input for calculating electrical conductivity and intrinsic carrier concentration. It's worth noting that typically, electrons are more mobile than holes due to their smaller effective mass.

Intrinsic carrier concentration relies on these mobilities because a high-mobility semiconductor material will require fewer charge carriers to achieve a particular level of conductivity. Conversely, semiconductor materials with lower mobilities will need a higher concentration of carriers to maintain the same conductivity.

Semiconductors are materials with electrical properties intermediate between conductors and insulators. They are the backbone of modern electronics, with applications in diodes, transistors, solar cells, and various other electronic devices.

PbS, which is mentioned in the exercise, is an example of a semiconductor material. The intrinsic carrier concentration, a vital parameter for these materials, significantly influences their electrical and optical properties.

As the exercise demonstrates, intrinsic carrier concentration can be calculated by using the electrical conductivity and the mobilities of electrons and holes. A higher intrinsic carrier concentration in a semiconductor corresponds to better conductivity. However, various external factors such as temperature, doping with other elements, and the manufacturing process can also greatly affect the behavior of these materials.

Understanding the basic properties of semiconductor materials like PbS allows us to precisely control their electrical characteristics by manipulating factors such as doping and carrier concentrations. This versatility makes semiconductors the building blocks of modern electronic technology.